Finding Numbers Divisible by 9: Applying the Divisibility Rule

The question asks us to identify which one of the given numbers is divisible by 9. To determine if a number is divisible by 9, we use a specific divisibility rule.

Understanding the Divisibility Rule for 9

A number is considered divisible by 9 if the sum of its digits is divisible by 9. This is a fundamental rule in number theory that helps us quickly check divisibility without performing long division.

Let's apply this rule to each of the given options:

Checking Each Number's Divisibility by 9

Option 1: 54321

We calculate the sum of the digits for this number:

Sum of digits = $5 + 4 + 3 + 2 + 1 = 15$.

Now, we check if this sum, 15, is divisible by 9. $15 \div 9$ does not result in a whole number. Therefore, 15 is not divisible by 9, which means the number 54321 is not divisible by 9.

Option 2: 87654

Next, we find the sum of the digits for 87654:

Sum of digits = $8 + 7 + 6 + 5 + 4 = 30$.

Is 30 divisible by 9? No, $30 \div 9$ is not a whole number. So, 30 is not divisible by 9, and consequently, the number 87654 is not divisible by 9.

Option 3: 56765

Let's calculate the sum of digits for 56765:

Sum of digits = $5 + 6 + 7 + 6 + 5 = 29$.

Is 29 divisible by 9? No, $29 \div 9$ is not a whole number. Therefore, 29 is not divisible by 9, which means the number 56765 is not divisible by 9.

Option 4: 47862

Finally, we calculate the sum of digits for the number 47862:

Sum of digits = $4 + 7 + 8 + 6 + 2 = 27$.

Is 27 divisible by 9? Yes, $27 \div 9 = 3$. Since the sum of the digits, 27, is divisible by 9, the number 47862 is divisible by 9.

Conclusion: Identifying the Number Divisible by 9

By applying the divisibility rule for 9 to each option, we found that only the sum of the digits of 47862 (which is 27) is divisible by 9. Therefore, 47862 is the number among the options that is divisible by 9.

Here is a summary of our findings:

Revision Table: Key Divisibility Rules

Understanding divisibility rules is essential for quick calculations in mathematics. Here's a quick look at some common divisibility rules:

Additional Information: Why the Divisibility Rule for 9 Works

The divisibility rule for 9 is not just a trick; it's based on the properties of numbers and place value. Any whole number can be expressed in expanded form using powers of 10. For instance, a three-digit number $abc$ can be written as $a \times 100 + b \times 10 + c \times 1$.

Consider the powers of 10 in relation to 9:

• $1 = 0 \times 9 + 1$
• $10 = 1 \times 9 + 1$
• $100 = 11 \times 9 + 1$
• $1000 = 111 \times 9 + 1$

In general, any power of 10 ($10^n$) leaves a remainder of 1 when divided by 9. We can write $10^n = \text{multiple of } 9 + 1$.

Now let's rewrite the expanded form of our number $abc$ using this idea:

$abc = a \times (100) + b \times (10) + c \times (1)$

$abc = a \times (\text{multiple of } 9 + 1) + b \times (\text{multiple of } 9 + 1) + c \times (\text{multiple of } 9 + 1)$

Expanding this:

$abc = (a \times \text{multiple of } 9 + a \times 1) + (b \times \text{multiple of } 9 + b \times 1) + (c \times \text{multiple of } 9 + c \times 1)$

$abc = (\text{multiple of } 9) + a + (\text{multiple of } 9) + b + (\text{multiple of } 9) + c$

$abc = (\text{sum of multiples of } 9) + (a + b + c)$

The sum of multiples of 9 is also a multiple of 9. So, the number $abc$ can be written as:

$abc = \text{a multiple of } 9 + (a + b + c)$

For $abc$ to be divisible by 9, the entire expression must be a multiple of 9. Since the first part (a multiple of 9) is already divisible by 9, the divisibility of $abc$ by 9 depends only on the divisibility of the second part, which is $(a + b + c)$, the sum of the digits. This is why the rule works!